Optimal. Leaf size=80 \[ -\frac{\sqrt{1-a x^2}}{2 a x^2 \sqrt{\frac{1}{a x^2+1}}}-\frac{1}{2 a x^2}-\frac{1}{2} \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} \sin ^{-1}\left (a x^2\right ) \]
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Rubi [A] time = 0.0530153, antiderivative size = 93, normalized size of antiderivative = 1.16, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {6334, 259, 275, 277, 216} \[ -\frac{\sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} \sqrt{1-a^2 x^4}}{2 a x^2}-\frac{1}{2 a x^2}-\frac{1}{2} \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} \sin ^{-1}\left (a x^2\right ) \]
Warning: Unable to verify antiderivative.
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Rule 6334
Rule 259
Rule 275
Rule 277
Rule 216
Rubi steps
\begin{align*} \int \frac{e^{\text{sech}^{-1}\left (a x^2\right )}}{x} \, dx &=-\frac{1}{2 a x^2}+\frac{\left (\sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{\sqrt{1-a x^2} \sqrt{1+a x^2}}{x^3} \, dx}{a}\\ &=-\frac{1}{2 a x^2}+\frac{\left (\sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{\sqrt{1-a^2 x^4}}{x^3} \, dx}{a}\\ &=-\frac{1}{2 a x^2}+\frac{\left (\sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-a^2 x^2}}{x^2} \, dx,x,x^2\right )}{2 a}\\ &=-\frac{1}{2 a x^2}-\frac{\sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2} \sqrt{1-a^2 x^4}}{2 a x^2}-\frac{1}{2} \left (a \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-a^2 x^2}} \, dx,x,x^2\right )\\ &=-\frac{1}{2 a x^2}-\frac{\sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2} \sqrt{1-a^2 x^4}}{2 a x^2}-\frac{1}{2} \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2} \sin ^{-1}\left (a x^2\right )\\ \end{align*}
Mathematica [A] time = 0.144055, size = 113, normalized size = 1.41 \[ \frac{\sqrt{\frac{1-a x^2}{a x^2+1}} \left (a x^2+1\right ) \tanh ^{-1}\left (\frac{a x^2}{\sqrt{a^2 x^4-1}}\right )}{2 \sqrt{a^2 x^4-1}}+\sqrt{\frac{1-a x^2}{a x^2+1}} \left (-\frac{1}{2 a x^2}-\frac{1}{2}\right )-\frac{1}{2 a x^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.284, size = 103, normalized size = 1.3 \begin{align*} -{\frac{1}{2}\sqrt{-{\frac{a{x}^{2}-1}{a{x}^{2}}}}\sqrt{{\frac{a{x}^{2}+1}{a{x}^{2}}}} \left ( \arctan \left ({{x}^{2}{\frac{1}{\sqrt{-{\frac{{a}^{2}{x}^{4}-1}{{a}^{2}}}}}}} \right ){x}^{2}+\sqrt{-{\frac{{a}^{2}{x}^{4}-1}{{a}^{2}}}} \right ){\frac{1}{\sqrt{-{\frac{{a}^{2}{x}^{4}-1}{{a}^{2}}}}}}}-{\frac{1}{2\,a{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{-\frac{a^{2} \arcsin \left (\frac{a^{2} x^{2}}{\sqrt{a^{2}}}\right )}{2 \, \sqrt{a^{2}}} - \frac{\sqrt{-a^{2} x^{4} + 1}}{2 \, x^{2}}}{a} - \frac{1}{2 \, a x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.0125, size = 230, normalized size = 2.88 \begin{align*} -\frac{a x^{2} \sqrt{\frac{a x^{2} + 1}{a x^{2}}} \sqrt{-\frac{a x^{2} - 1}{a x^{2}}} - 2 \, a x^{2} \arctan \left (\frac{a x^{2} \sqrt{\frac{a x^{2} + 1}{a x^{2}}} \sqrt{-\frac{a x^{2} - 1}{a x^{2}}} - 1}{a x^{2}}\right ) + 1}{2 \, a x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{x^{3}}\, dx + \int \frac{a \sqrt{-1 + \frac{1}{a x^{2}}} \sqrt{1 + \frac{1}{a x^{2}}}}{x}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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